Alexander, K. S.
The Asymmetric Random Cluster Model and Comparison of Ising and Potts Models
(162K, AMS-LATeX 1.2)
ABSTRACT. We introduce the asymmetric random cluster (or ARC) model, which is a
graphical representation of the Potts lattice gas, and establish its basic
properties. The ARC model allows a rich variety of comparisons (in the FKG
sense) between models with different parameter values; we give, for example,
values $(\beta, h)$ for which the 0's configuration in the Potts lattice gas
is dominated by the ``+'' configuration of the $(\beta,h)$ Ising model.
The Potts model, with possibly an external field applied to one of the spins,
is a special case of the Potts lattice gas, which allows our comparisons
to yield rigorous bounds on the critical temperatures of Potts models. For
example, we obtain $.571 \leq 1 - \exp(-\beta_{c}) \leq .600$ for the 9-state
Potts model on the hexagonal lattice. Another comparison bounds the movement
of the critical line when a small Potts interaction is added to a lattice gas
which otherwise has only interparticle attraction. ARC models can also be
compared to related models such as the partial FK model, obtained by
deleting a fraction of the nonsingleton clusters from a realization of
the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds
on the effects of small annealed site dilution on the critical temperature of
the Potts model.